5.1.2 Transform definitions and Storage for Real Data

The DFT of a sequence of real data results in a special form of complex sequence known as a Hermitian sequence. The symmetries defining such a sequence mean that it can be fully represented by a set of n real values, where n is the length of the original real sequence. It is therefore conventional for the array containing the real sequence to be overwritten by such a representation of the transformed Hermitian sequence.

In full complex representation, the Hermitian sequence would be a sequence of n complex values Z(i) for i=0,1,...,n-1, where Z(n-j) is the complex conjugate of Z(j) for j=1,2,...,(n-1)/2; Z(0) is real valued; and, if n is even, Z(n/2) is real valued. In ACML, the representation of Hermitian sequences used on output from DZFFT routines and on input to ZDFFT routines is as follows:
let X be an array of length N and with first index 0,

• X(i) contains the real part of Z(i) for i=0,...,N/2
• X(N-i) contains the imaginary part of Z(i) for i=1,...,(N-1)/2